Randomized Distributed Decision

Abstract

The paper tackles the power of randomization in the context of locality by analyzing the ability to “boost” the success probability of deciding a distributed language. The main outcome of this analysis is that the distributed computing setting contrasts significantly with the sequential one as far as randomization is concerned. Indeed, we prove that in some cases, the ability to increase the success probability for deciding distributed languages is rather limited.

We focus on the notion of a (p,q)-decider for a language $\mathcal{L}$, which is a distributed randomized algorithm that accepts instances in $\mathcal{L}$ with probability at least p and rejects instances outside of $\mathcal{L}$ with probability at least q. It is known that every hereditary language that can be decided in t rounds by a (p,q)-decider, where p ² + q > 1, can be decided deterministically in O(t) rounds. One of our results gives evidence supporting the conjecture that the above statement holds for all distributed languages and not only for hereditary ones, by proving the conjecture for the restricted case of path topologies.

For the range below the aforementioned threshold, namely, p ² + q ≤ 1, we study the class B _k (t) (for k ∈ ℕ^* ∪ { ∞ }) of all languages decidable in at most t rounds by a (p,q)-decider, where $p^{1+\frac{1}{k}}+q>1$. Since every language is decidable (in zero rounds) by a (p,q)-decider satisfying p + q = 1, the hierarchy B _k provides a spectrum of complexity classes between determinism (k = 1, under the above conjecture) and complete randomization (k = ∞). We prove that all these classes are separated, in a strong sense: for every integer k ≥ 1, there exists a language $\mathcal{L}$ satisfying $\mathcal{L}\in B_{k+1}(0)$ but $\mathcal{L}\notin B_k(t)$ for any t = o(n). In addition, we show that B _∞(t) does not contain all languages, for any t = o(n). In other words, we obtain the hierarchy B ₁(t) ⊂ B ₂ (t) ⊂ ⋯ ⊂ B _∞(t) ⊂ All.

Finally, we show that if the inputs can be restricted in certain ways, then the ability to boost the success probability becomes almost null, and in particular, derandomization is not possible even beyond the threshold p ² + q = 1.